1. Kinematic Modelling in Robotics
dr Dragan Kostić WTB Dynamics and Control
October 22th, 2010

2. Outline Representing rotations and rotational transformations
Parameterization of rotations
Rigid motions and homogenous transformations
DH convention for modeling of robot kinematics
Forward kinematics
Case-study: kinematics of RRR-arm

3. Representing rotations in coordinate frame 0
Rotation matrix
xi and yi are the unit vectors in oixiyi

4. Representing rotations in coordinate frame 1

5. Representing rotations in 3D (1/4)
Each axis of the frame o1x1y1z1 is projected onto o0x0y0z0:
R10  SO(3)

6. Representing rotations in 3D (2/4)
Example: Frame o1x1y1z1 is obtained from frame o0x0y0z by rotation through an angle  about z0 axis.
all other dot products are zero

7. Representing rotations in 3D (3/4)
Basic rotation matrix about z-axis

8. Representing rotations in 3D (4/4)
Similarly, basic rotation matrices about x- and y-axes:

9. Rotational transformations
pi: coordinates of p in oixiyizi

10. Parameterization of rotations (1/2)
Euler angles
ZYZEuler angle transformation:

11. Parameterization of rotations (2/2)
Roll, pitch, yaw angles
XYZyaw-pitch-roll angle transformation:

12. Rigid motions

13. Homogenous transformations (1/2)
We have
Note that
Consequently, rigid motion (d, R) can be described by matrix representing homogenous transformation:

14. Homogenous transformations (2/2)
Since R is orthogonal, we have
We augment vectors p0 and p1 to get their homogenous representations
and achieve matrix representation of coordinate transformation

15. Basic homogenous transformations

16. Conventions (1/2)
1. there are n joints and hence n + 1 links; joints 1, 2, , n; links 0, 1, , n,
2. joint i connects link i − 1 to link i,
3. actuation of joint i causes link i to move,
4. link 0 (the base) is fixed and does not move,
5. each joint has a single degree-of-freedom (dof):

17. Conventions (2/2)
6. frame oixiyizi is attached to link i; regardless of motion of the robot, coordinates of each point on link i are constant when expressed in frame oixiyizi,
7. when joint i is actuated, link i and its attached frame oixiyizi experience resulting motion.

18. DH convention for homogenous transformations
Position and orientation of coordinate frame i with respect to frame i-1 is specified by homogenous transformation matrix:
where

19. Physical meaning of DH parameters
Link length ai is distance from zi-1 to zi measured along xi.
Link twist i is angle between zi-1 and zi measured in plane normal to xi (right-hand rule).
Link offset di is distance from origin of frame i-1 to the intersection xi with zi-1, measured along zi-1.
Joint angle i is angle from xi-1 to xi measured in plane normal to zi-1 (right-hand rule).

20. DH convention to assign coordinate frames
Assign zi to be the axis of actuation for joint i+1 (unless otherwise stated zn coincides with zn-1).
Choose x0 and y0 so that the base frame is right-handed.
Iterative procedure for choosing oixiyizi depending on oi-1xi-1yi-1zi-1 (i=1, 2, , n-1):
a) zi−1 and zi are not coplanar; there is an unique shortest line segment from zi−1 to zi, perpendicular to both; this line segment defines xi and the point where the line intersects zi is the origin oi; choose yi to form a right-handed frame,
b) zi−1 is parallel to zi; there are infinitely many common normals; choose xi as the normal passes through oi−1; choose oi as the point at which this normal intersects zi; choose yi to form a right-handed frame,
c) zi−1 intersects zi; axis xi is chosen normal to the plane formed by zi and zi−1; it’s positive direction is arbitrary; the most natural choice of oi is the intersection of zi and zi−1, however, any point along the zi suffices; choose yi to form a right-handed frame.

21. Forward kinematics (1/2)
Homogenous transformation matrix relating the frame oixiyizi to oi-1xi-1yi-1zi-1:
Ai specifies position and orientation of oixiyizi w.r.t. oi-1xi-1yi-1zi-1.
Homogenous transformation matrix Tji expresses position and orientation of ojxjyjzj with respect to oixiyizi:

22. Forward kinematics (2/2)
Forward kinematics of a serial manipulator with n joints can be represented by homogenous transformation matrix Hn0 which defines position and orientation of the end-effector’s (tip) frame onxnynzn relative to the base coordinate frame o0x0y0z0:

23. Case-study: RRR robot manipulator

24. DH parameters of RRR robot manipulator

25. Forward kinematics of RRR robot manipulator (1/2)
Coordinate frame o3x3y3z3 is related with the base frame o0x0y0z0 via homogenous transformation matrix:
where

26. Forward kinematics of RRR robot manipulator (2/2)
Position of end-effector: , ,
Orientation of end-effector: